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We study theoretically the low-energy hole states in Si, Ge, and Ge/Sicore/shell nanowires (NWs). The NW core in our model has a rectangular crosssection, the results for a square cross section are presented in detail. In thecase of Ge and Ge/Si core/shell NWs, we obtain very good agreement withprevious theoretical results for cylindrically symmetric NWs. In particular,the NWs allow for an unusually strong and electrically controllable spin-orbitinteraction (SOI) of Rashba type. We find that the dominant contribution to theSOI is the "direct Rashba spin-orbit interaction" (DRSOI), which is animportant mechanism for systems with heavy-hole-light-hole mixing. Our resultsfor Si NWs depend significantly on the orientation of the crystallographicaxes. The numerically observed dependence on the growth direction is consistentwith analytical results from a simple model, and we identify a setup where theDRSOI enables spin-orbit energies of the order of millielectronvolts in Si NWs.Furthermore, we analyze the dependence of the SOI on the electric field and thecross section of the Ge or Si core. A helical gap in the spectrum can be openedwith a magnetic field. We obtain the largest g factors with magnetic fieldsapplied perpendicularly to the NWs.

Direct Rashba spin-orbit interaction in Si and Ge nanowires with different growth directions